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Monte-Carlo Planning II

Monte-Carlo Planning II. Alan Fern. Outline. Preliminaries: Markov Decision Processes What is Monte-Carlo Planning? Uniform Monte-Carlo Single State Case (UniformBandit) Policy rollout Approximate Policy Iteration Sparse Sampling Adaptive Monte-Carlo Single State Case (UCB Bandit)

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Monte-Carlo Planning II

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  1. Monte-Carlo Planning II Alan Fern

  2. Outline • Preliminaries: Markov Decision Processes • What is Monte-Carlo Planning? • Uniform Monte-Carlo • Single State Case (UniformBandit) • Policy rollout • Approximate Policy Iteration • Sparse Sampling • Adaptive Monte-Carlo • Single State Case (UCB Bandit) • UCT Monte-Carlo Tree Search

  3. Sparse Sampling • Rollout does not guarantee optimality or near optimality • Neither does approximate policy iteration in general • Can we develop Monte-Carlo methods that give us near optimal policies? • With computation that does NOT depend on number of states! • This was an open problem until late 90’s. • In deterministic games and search problems it is common to build a look-ahead tree at a state to select best action • Can we generalize this to general stochastic MDPs? • Sparse Sampling is one such algorithm • Strong theoretical guarantees of near optimality

  4. Online Planning with Look-Ahead Trees • At each state we encounter in the environment we build a look-ahead tree of depth hand use it to estimate optimal Q-values of each action • Select action with highest Q-value estimate • s = current state • Repeat • T = BuildLookAheadTree(s) ;; sparse sampling or UCT ;; tree provides Q-value estimates for root action • a = BestRootAction(T) ;; action with best Q-value • Execute action a in environment • s is the resulting state

  5. Sparse Sampling • Again focus on finite-horizons • Arbitrarily good approximation for large enough horizon h • h-horizon optimal Q-function • Value of taking a in s and following for * for h-1 steps • Q*(s,a,h) = E[R(s,a) + βV*(T(s,a),h-1)] • Key identity (Bellman’s equations): • V*(s,h) = maxaQ*(s,a,h) • *(x) = argmaxaQ*(x,a,h) • Sparse sampling estimates Q-values by building sparse expectimax tree

  6. Sparse Sampling • Will present two views of algorithm • The first is perhaps easier to digest • The second is more generalizable and can leverage advances in bandit algorithms • Approximation to the full expectimax tree • Recursive bandit algorithm

  7. Expectimax Tree • Key definitions: • V*(s,h) = maxaQ*(s,a,h) • Q*(s,a,h) = E[R(s,a) + βV*(T(s,a),h-1)] • Expand definitions recursively to compute V*(s,h) V*(s,h) = maxa1Q(s,a1,h) = maxa1E[R(s,a1) + β V*(T(s,a1),h-1)] = maxa1E[R(s,a1) + β maxa2E[R(T(s,a1),a2)+Q*(T(s,a1),a2,h-1)] = …… • Can view this expansion as an expectimax tree • Each expectation is really a weighted sum over states

  8. Exact Expectimax Tree for V*(s,H) V*(s,H) Alternate max &expectation Q*(s,a,H) Compute root V* and Q* via recursive procedure Depends on size of the state-space. Bad!

  9. Sparse Sampling Tree V*(s,H) Q*(s,a,H) Sampling width w (kw)H leaves Replace expectation with average over w samples w will typically be much smaller than n.

  10. Sparse Sampling [Kearns et. al. 2002] The Sparse Sampling algorithm computes root value via depth first expansion Return value estimate V*(s,h) of state s and estimated optimal action a* SparseSampleTree(s,h,w) For each action a in s Q*(s,a,h) = 0 For i = 1 to w Simulate taking a in s resulting in si and reward ri [V*(si,h),a*] = SparseSample(si,h-1,w) Q*(s,a,h) = Q*(s,a,h) + ri+ β V*(si,h) Q*(s,a,h) = Q*(s,a,h) / w ;; estimate of Q*(s,a,h) V*(s,h) = maxa Q*(s,a,h) ;; estimate of V*(s,h) a* = argmaxa Q*(s,a,h) Return [V*(s,h), a*]

  11. Sparse Sampling (Cont’d) • For a given desired accuracy, how largeshould sampling width and depth be? • Answered: Kearns, Mansour, and Ng (1999) • Good news: gives values for w and H to achieve policy arbitrarily close to optimal • Values are independent of state-space size! • First near-optimal general MDP planning algorithm whose runtime didn’t depend on size of state-space • Bad news: the theoretical values are typically still intractably large---also exponential in H • Exponential in H is the best we can do in general • In practice: use small Hand use heuristic at leaves

  12. Sparse Sampling • Will present two views of algorithm • The first is perhaps easier to digest • The second is more generalizable and can leverage advances in bandit algorithms • Approximation to the full expectimax tree • Recursive bandit algorithm

  13. Bandit View of Expectimax Tree V*(s,H) Alternate maxand expectaton Q*(s,a,H) Each max node in tree is just a bandit problem. I.e. must choose action with highest Q*(s,a,h)---approximate via bandit.

  14. Bandit View Assuming we Know V* s SimQ*(s,ai,h) = R(s,ai) + β V*(T(s,ai),h-1) ak a1 a2 … SimQ*(s,a1,h) SimQ*(s,a2,h) SimQ*(s,ak,h) • SimQ*(s,a,h) • s’ = T(s,a) • r = R(s,a) • Return r + β V*(s’,h-1) • Expected value of SimQ*(s,a,h) is Q*(s,a,h) • Use UniformBandit to select approximately optimal action

  15. But we don’t know V* • To compute SimQ*(s,a,h) need V*(s’,h-1) for any s’ • Use recursive identity (Bellman’s equation): • V*(s,h-1) = maxaQ*(s,a,h-1) • Idea: Can recursively estimate V*(s,h-1) by running h-1 horizon bandit based on SimQ* • Bandit returns estimated value of best action rather than just returning best action • Base Case: V*(s,0) = 0, for all s

  16. Recursive UniformBandit s ak a1 a2 SimQ(s,ai,h) Recursively generate samples of R(s,ai) + β V*(T(s,ai),h-1) … … q1w SimQ*(s,a2,h) SimQ*(s,ak,h) q11 q12 UniformBandit will generate w sample next states from each action at each state

  17. Recursive UniformBandit s ak a1 a2 SimQ(s,ai,h) Recursively generate samples of R(s,ai) + β V*(T(s,ai),h-1) … SimQ*(s,a1,h) … q1w SimQ*(s,a2,h) SimQ*(s,ak,h) q11 q12 Returns V*(s11,h-1)estimate Returns V*(s12,h-1)estimate s11 s12 a1 a1 ak ak … … … … SimQ*(s12,ak,h-1) SimQ*(s12,a1,h-1) SimQ*(s11,a1,h-1) SimQ*(s11,ak,h-1)

  18. Bandit View s ak a1 a2 … … q1w SimQ*(s,a2,h) SimQ*(s,ak,h) q11 q12 • When bandit is UniformBandit same as sparse sampling • Each state generates kw new states (w states for each of k bandits) • Total # of states in tree (kw)h • Can plug in more advanced bandit algorithms for possible improvement! s11 a1 ak … …

  19. Uniform vs. Adaptive Bandits • Sparse sampling wastes time on bad parts of tree • Devotes equal resources to each state encountered in the tree • Would like to focus on most promising parts of tree • But how to control exploration of new parts of tree vs. exploiting promising parts? • Need adaptive bandit algorithmthat explores more effectively

  20. Outline • Preliminaries: Markov Decision Processes • What is Monte-Carlo Planning? • Uniform Monte-Carlo • Single State Case (UniformBandit) • Policy rollout • Sparse Sampling • Adaptive Monte-Carlo • Single State Case (UCB Bandit) • UCT Monte-Carlo Tree Search

  21. Regret Minimization Bandit Objective • Problem: find arm-pulling strategy such that the expected total reward at time n is close to the best possible (one pull per time step) • Optimal (in expectation) is to pull optimal arm n times • UniformBandit is poor choice --- waste time on bad arms • Must balance exploring machines to find good payoffs and exploiting current knowledge s ak a1 a2 …

  22. UCB Adaptive Bandit Algorithm[Auer, Cesa-Bianchi, & Fischer, 2002] • Q(a) : average payoff for action a (in our single state s) based on current experience • n(a) : number of pulls of arm a • Action choice by UCB after n pulls: • Theorem: The expected regret (i.e. sub-optimality) after n arm pulls compared to optimal behavior is bounded by O(log n) • No algorithm can achieve a better loss rate Assumes payoffs in [0,1]

  23. UCB Algorithm [Auer, Cesa-Bianchi, & Fischer, 2002] Value Term:favors actions that looked good historically Exploration Term:actions get an exploration bonus that grows with ln(n) Expected number of pulls of sub-optimal arm a is bounded by: where is regret of arm a (i.e. the amount of sub-optimality) Doesn’t waste much time on sub-optimal arms, unlike uniform!

  24. UCB for Multi-State MDPs • UCB-Based Policy Rollout: • Use UCB to select actions instead of uniform • Likely to use samples more efficiently by concentrating on promising actions • Unclear if this could be shown to have PAC properties • UCB-Based Sparse Sampling • Use UCB to make sampling decisions at internal tree nodes • There is an analysis of this algorithm’s bias H.S. Chang, M. Fu, J. Hu, and S.I. Marcus. An adaptive sampling algorithm for solving Markov decision processes. Operations Research, 53(1):126--139, 2005.

  25. UCB-based Sparse Sampling [Chang et. al. 2005] s • Use UCB instead of Uniform to direct sampling at each state • Non-uniform allocation ak a1 a2 … q11 q21 q22 q31 q32 s11 s11 a1 ak … • But each qijsample requires waiting for an entire recursive h-1 level tree search • Better, but still very expensive! … SimQ*(s11,a1,h-1) SimQ*(s11,ak,h-1)

  26. Outline • Preliminaries: Markov Decision Processes • What is Monte-Carlo Planning? • Uniform Monte-Carlo • Single State Case (UniformBandit) • Policy rollout • Sparse Sampling • Adaptive Monte-Carlo • Single State Case (UCB Bandit) • UCT Monte-Carlo Tree Search

  27. UCT Algorithm [Kocsis & Szepesvari, 2006] • UCT is an instance of Monte-Carlo Tree Search • Applies principle of UCB • Similar theoretical properties to sparse sampling • Much better anytime behavior than sparse sampling • Famous for yielding a major advance in computer Go • A growing number of success stories • Practical successes still not fully understood

  28. Monte-Carlo Tree Search What is the rollout policy? • Builds a sparse look-ahead tree rooted at current state by repeated Monte-Carlo simulation of a “rollout policy” • During construction each tree node s stores: • state-visitation count n(s) • action counts n(s,a) • action values Q(s,a) • Repeat until time is up • Execute rollout policy starting from root until horizon(generates a state-action-reward trajectory) • Add first node not in current tree to the tree • Update statistics of each tree nodes on trajectory • Increment n(s) and n(s,a) for selected action a • Update Q(s,a) by total reward observed after the node

  29. Rollout Policies • Monte-Carlo Tree Search algorithms mainly differ on their choice of rollout policy • Rollout policies have two distinct phases • Tree policy: selects actions at nodes already in tree(each action must be selected at least once) • Default policy: selects actions after leaving tree • Key Idea: the tree policy can use statistics collected from previous trajectories to intelligently expand tree in most promising direction • Rather than uniformly explore actions at each node

  30. At a leaf node tree policy selects a random action then executes default Iteration 1 Current World State Initially tree is single leaf 1 1 new tree node DefaultPolicy 1 Terminal(reward = 1) Assume all non-zero reward occurs at terminal nodes.

  31. Must select each action at a node at least once Iteration 2 Current World State 1 1 new tree node DefaultPolicy 0 Terminal(reward = 0)

  32. Must select each action at a node at least once Iteration 3 Current World State 1/2 1 0

  33. When all node actions tried once, select action according to tree policy Iteration 3 Current World State 1/2 Tree Policy 1 0

  34. When all node actions tried once, select action according to tree policy Iteration 3 Current World State 1/2 Tree Policy 1 0 new tree node DefaultPolicy 0

  35. When all node actions tried once, select action according to tree policy Iteration 4 Current World State 1/3 Tree Policy 0 1/2 0

  36. When all node actions tried once, select action according to tree policy Iteration 4 Current World State 1/3 0 1/2 0 1

  37. When all node actions tried once, select action according to tree policy Current World State 2/3 Tree Policy 0 2/3 0 1 What is an appropriate tree policy?Default policy?

  38. UCT Algorithm [Kocsis & Szepesvari, 2006] • Basic UCT uses random default policy • In practice often use hand-coded or learned policy • Tree policy is based on UCB: • Q(s,a) : average reward received in current trajectories after taking action a in state s • n(s,a) : number of times action a taken in s • n(s) : number of times state s encountered Theoretical constant that must be selected empirically in practice

  39. When all node actions tried once, select action according to tree policy Current World State 1/3 a2 a1 Tree Policy 0 1/2 0 1

  40. When all node actions tried once, select action according to tree policy Current World State 1/3 Tree Policy 0 1/2 0 1

  41. When all node actions tried once, select action according to tree policy Current World State 1/3 Tree Policy 0 1/2 0 1 0

  42. When all node actions tried once, select action according to tree policy Current World State 1/4 Tree Policy 0 1/3 1 0/1 0

  43. When all node actions tried once, select action according to tree policy Current World State 1/4 Tree Policy 0 1/3 1 0/1 0 1

  44. When all node actions tried once, select action according to tree policy Current World State 2/5 Tree Policy 1/2 1/3 1 1 0/1 0

  45. UCT Recap • To select an action at a state s • Build a tree using N iterations of monte-carlo tree search • Default policy is uniform random • Tree policy is based on UCB rule • Select action that maximizes Q(s,a)(note that this final action selection does not take the exploration term into account, just the Q-value estimate) • The more simulations the more accurate

  46. Computer Go • “Task Par Excellence for AI” (Hans Berliner) • “New Drosophila of AI” (John McCarthy) • “Grand Challenge Task” (David Mechner) 9x9 (smallest board) 19x19 (largest board)

  47. A Brief History of Computer Go • 2005: Computer Go is impossible! • 2006: UCT invented and applied to 9x9 Go (Kocsis, Szepesvari; Gelly et al.) • 2007: Human master level achieved at 9x9 Go (Gelly, Silver; Coulom) • 2008: Human grandmaster level achieved at 9x9 Go (Teytaud et al.) Computer GO Server rating over this period: 1800 ELO  2600 ELO

  48. Other Successes • Klondike Solitaire (wins 40% of games) • General Game Playing Competition • Real-Time Strategy Games • Combinatorial Optimization • List is growing • Usually extend UCT is some ways

  49. Some Improvements • Use domain knowledge to handcraft a more intelligent default policy than random • E.g. don’t choose obviously stupid actions • In Go a hand-coded default policy is used • Learn a heuristic function to evaluate positions • Use the heuristic function to initialize leaf nodes(otherwise initialized to zero)

  50. Summary • When you have a tough planning problem and a simulator • Try Monte-Carlo planning • Basic principles derive from the multi-arm bandit • Policy Rollout is a great way to exploit existing policies and make them better • If a good heuristic exists, then shallow sparse sampling can give good gains • UCT is often quite effective especially when combined with domain knowledge

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